CHAPTER PPREREQUISITES1
P.1 ModelingtheRealWorldwithAlgebra1
P.2 RealNumbers2
P.3 IntegerExponentsandScientificNotation7
P.4 RationalExponentsandRadicals12
P.5 AlgebraicExpressions16
P.6 Factoring19
P.7 RationalExpressions24
P.8 SolvingBasicEquations31
P.9 ModelingwithEquations36 ChapterPReview42 ChapterPTest48
¥ FOCUSONMODELING: MakingOptimalDecisions51
PREREQUISITES
P.1
MODELINGTHEREALWORLDWITHALGEBRA
1. Usingthismodel,wefindthat15carshave W 4 15 60wheels.Tofindthenumberofcarsthathaveatotalof W wheels,wewrite W 4 X X W 4 .Ifthecarsinaparkinglothaveatotalof124wheels,wefindthatthereare X 124 4 31carsinthelot.
2. Ifeachgallonofgascosts$350,then x gallonsofgascosts$35 x .Thus, C 35 x .Wefindthat12gallonsofgaswould cost C 3 5 12 $42.
3. If x $120and T 0 06 x ,then T 0 06 120 7 2.Thesalestaxis$7 20.
4. If x 62,000and T 0 005 x ,then T 0 005 62,000 310.Thewagetaxis$310.
5. If 70, t 3 5,and d t ,then d 70 3 5 245.Thecarhastraveled245miles.
6. V r 2 h 32 5 45 141 4in3
7.(a) M N G 240 8 30miles/gallon (b) 25 175 G G 175 25 7gallons 8.(a) T 70 0
11.(a)
12.(a)
(b) Weknowthat P 30andwewanttofind d ,sowesolvethe equation30 14 7
45 340.Thus,ifthepressureis30lb/in2 ,thedepth is34ft.
(b) Wesolvetheequation40 x 120,000 x 120,000 40 3000.Thus,thepopulationisabout3000.
13. Thenumber N ofcentsin q quartersis N 25q
14. Theaverage A
15. Thecost C ofpurchasing x gallonsofgasat$3 50agallonis C 3 5 x
16. Theamount T ofa15%tiponarestaurantbillof x dollarsis T 0 15 x
17. Thedistance d inmilesthatacartravelsin t hoursat60mi/his d 60t
18. Thespeed r ofaboatthattravels d milesin3hoursis r d 3
19.(a) $12 3
$1 $12 $3 $15
(b) Thecost C ,indollars,ofapizzawith n toppingsis C 12 n
(c) Usingthemodel C 12 n with C 16,weget16 12 n n 4.Sothepizzahasfourtoppings.
20.(a) 3
30
$118
(b) Thecostis daily rental days rented
cost permile
miles driven
(c) Wehave C 140and n 3.Substituting,weget140
m 500.Sotherentalwasdriven500miles.
21.(a)(i) Foranallelectriccar,theenergycostofdriving x milesis C e 0 04 x (ii) Foranaveragegasolinepoweredcar,theenergycostofdriving x milesis C g 0 12 x (b)(i) Thecostofdriving10,000mileswithanallelectriccaris C
(ii) Thecostofdriving10,000mileswithagasolinepoweredcaris
22.(a) Ifthewidthis20,thenthelengthis40,sothevolumeis20 20 40 16,000in3 (b) Intermsofwidth, V x
23.(a) TheGPAis
(b) Using a 2 3 6, b 4, c 3 3 9,and
P.2 THEREALNUMBERS
1.(a) Thenaturalnumbersare 1
2
3
(b) Thenumbers
f 0intheformulafrompart(a),wefindtheGPAtobe
areintegersbutnotnaturalnumbers.
(c) Anyirreduciblefraction p q with q 1isrationalbutisnotaninteger.Examples: 3 2 , 5 12 , 1729 23
(d) Anynumberwhichcannotbeexpressedasaratio p q oftwointegersisirrational.Examplesare 2, 3, ,and e
2.(a) ab ba ;CommutativePropertyofMultiplication
(b) a b c a b c ;AssociativePropertyofAddition
(c) a b c ab ac;DistributiveProperty
3.(a) Insetbuildernotation: x 3 x 5
(b) Inintervalnotation: 3 5
(c) Asagraph: _35
4. Thesymbol x standsforthe absolutevalue ofthenumber x .If x isnot0,thenthesignof x isalways positive
5. Thedistancebetween a and b onthereallineis d a
b a
.Sothedistancebetween 5and2is
6.(a) If a b ,thenanyintervalbetween a and b (whetherornotitcontainseitherendpoint)containsinfinitelymany numbers—including,forexample a b a 2n foreverypositive n .(Ifanintervalextendstoinfinityineitherorboth directions,thenitobviouslycontainsinfinitelymanynumbers.)
(b) No,because 5 6 doesnotinclude5.
7.(a) No: a b b a b a ingeneral.
(b) No;bytheDistributiveProperty, 2 a
8.(a) Yes,absolutevalues(suchasthedistancebetweentwodifferentnumbers) arealwayspositive.
(b) Yes, b a a b
9.(a) Naturalnumber:100
(b) Integers:0,100, 8
(c) Rationalnumbers: 1 5,0, 5 2 ,2 71,3 14,100, 8
(d) Irrationalnumbers: 7,
11. CommutativePropertyofaddition
Naturalnumbers:2, 9 3,10
(b) Integers:2, 100 2 50,
(c) Rationalnumbers:4
(d) Irrationalnumbers: 2, 3 14
CommutativePropertyofmultiplication 13. AssociativePropertyofaddition
DistributiveProperty
CommutativePropertyofmultiplication
83.(a) a isnegativebecause a ispositive.
(b) bc ispositivebecausetheproductoftwonegativenumbersispositive.
(c) a ba b ispositivebecauseitisthesumoftwopositivenumbers.
(d) ab ac isnegative:eachsummandistheproductofapositivenumberandanegative number,andthesumoftwo negativenumbersisnegative.
84.(a) b ispositivebecause b isnegative.
(b) a bc ispositivebecauseitisthesumoftwopositivenumbers.
(c) c a c a isnegativebecause c and a arebothnegative.
(d) ab2 ispositivebecauseboth a and b 2 arepositive.
85. DistributiveProperty
86.(a) When L 60, x 8,and y 6,wehave L
60 28 88.Because88 108the postofficewillacceptthispackage. When L 48, x 24,and y 24,wehave
48 96 144,andsince 144 108,thepostofficewill not acceptthispackage. (b) If x y 9,then
72.Sothelengthcanbeaslongas72in. 6ft.
87. Let x m 1 n 1 and y m 2 n 2 berationalnumbers.Then
,and
.Thisshowsthatthesum,difference,andproduct oftworationalnumbersareagainrationalnumbers.Howevertheproductof twoirrationalnumbersisnotnecessarily irrational;forexample, 2 2 2,whichisrational.Also,thesumoftwoirrationalnumbersisnotnecessarilyirrational; forexample, 2 2 0whichisrational.
88. 1 2 2isirrational.Ifitwererational,thenbyExercise6(a),thesum 1 2
2wouldberational,but thisisnotthecase.
Similarly, 1 2 2isirrational.
(a) Followingthehint,supposethat r t q ,arationalnumber.ThenbyExercise6(a),thesumofthetworational numbers r t and r isrational.But r t r t ,whichweknowtobeirrational.Thisisacontradiction,and henceouroriginalpremise—that r t isrational—wasfalse.
(b) r isanonzerorationalnumber,so r a b forsomenonzerointegers a and b .Letusassumethat rt q ,arational number.Thenbydefinition, q c d forsomeintegers c and d .Butthen rt q a b t c d ,whence t bc ad ,implying that t isrational.Onceagainwehavearrivedatacontradiction,andweconclude thattheproductofarationalnumber andanirrationalnumberisirrational.
89.
As x getslarge,thefraction1 x getssmall.Mathematically,wesaythat1 x goestozero.
As x getssmall,thefraction1 x getslarge.Mathematically,wesaythat1 x goestoinfinity.
90. Wecanconstructthenumber 2onthenumberlineby transferringthelengthofthe hypotenuseofarighttriangle withlegsoflength1and1. Similarly,tolocate 5,weconstructarighttrianglewithlegs oflength1and2.BythePythagoreanTheorem,thelength ofthehypotenuseis 12 22 5.Thentransferthe lengthofthehypotenusetothenumberline. Thesquarerootofanyrationalnumbercanbelocatedona numberlineinthisfashion.
Thecircleinthesecondfigureinthetexthascircumference ,soifwerollitalonganumberlineonefullrotation,wehave found onthenumberline.Similarly,anyrationalmultipleof canbefoundthisway.
91.(a) Supposethat a b ,somax
Ontheotherhand,if b
If a b ,then
(b) If a b ,thenmin
Similarly,if b a ,then
92. Answerswillvary.
0andtheresultistrivial.
and
,theresultistrivial.
93.(a) Subtractionisnotcommutative.Forexample,5 1 1 5.
(b) Divisionisnotcommutative.Forexample,5 1 1 5.
(c) Puttingonyoursocksandputtingonyourshoesarenotcommutative.Ifyouputonyoursocksfirst,thenyourshoes, theresultisnotthesameasifyouproceedtheotherwayaround.
(d) Puttingonyourhatandputtingonyourcoatarecommutative.Theycanbedoneineitherorder,withthesameresult.
(e) Washinglaundryanddryingitarenotcommutative.
94.(a) If x 2and y 3,then x
If x 2and y 3,then
If x 2and y 3,then
Ineachcase, x y
y andtheTriangleInequalityissatisfied.
(b) Case0: Ifeither x or y is0,theresultisequality,trivially.
Case1: If x and y havethesamesign,then
x y
y if x and y arepositive
if x and y arenegative
Case2: If x and y haveoppositesigns,thensupposewithoutlossofgeneralitythat x 0and y 0.Then
P.3 INTEGEREXPONENTSANDSCIENTIFICNOTATION
1. Usingexponentialnotationwecanwritetheproduct5
2. Yes,thereisadifference:
3. Intheexpression34 ,thenumber3iscalledthe base andthenumber4iscalledthe exponent 4.(a) Whenwemultiplytwopowerswiththesamebase,we add theexponents.So34 35 39 (b) Whenwedividetwopowerswiththesamebase,we subtract theexponents.So 35 32 33
5. Whenweraiseapowertoanewpower,we multiply theexponents.So
6.(a) 2
38
7. Tomoveanumberraisedtoapowerfromnumeratortodenominatororfromdenominatortonumeratorchangethesignof the
8. Scientistsexpressverylargeorverysmallnumbersusing scientific notation.Inscientificnotation,8,300,000is8 3 106 and0 0000327is3 27 10 5
b
(c) 2 y 10 y 11 2 y 10
21.(a) x 5 x 3 x 53 x 2 1 x 2 (b) 2
32.(a) 5 xy 2 x 1 y 3 5
33.(a) 69,300,000 6 93 107 (b) 7,200,000,000,000 7 2 1012
(c) 0 000028536 2 8536 10 5
(d) 0 0001213 1 213 10 4
35.(a) 319 105 319,000 (b) 2721 108 272,100,000
(c) 2670 10 8 000000002670
(d) 9 999 10 9 0 000000009999
37.(a) 5,900,000,000,000mi 5 9 1012 mi
34.(a) 129,540,000 1 2954 108 (b) 7,259,000,000 7 259 109
(c) 0 0000000014 1 4 10 9 (d) 0 0007029 7 029 10 4
36.(a) 71 1014 710,000,000,000,000 (b) 6 1012 6,000,000,000,000
(c) 855 10 3 000855 (d) 6 257 10 10 0 0000000006257
(b)
38.(a) 93,000,000mi
(b)
39.
40.
10100 isto10101
46.(a) b 5 isnegativesinceanegativenumberraisedtoanoddpowerisnegative.
(b) b 10 ispositivesinceanegativenumberraisedtoanevenpowerispositive.
(c) ab2 c3 wehave positive
(d) Since b a isnegative, b a 3 negative3 whichisnegative.
(e) Since b a isnegative, b a 4 negative4 whichispositive.
(f) a 3 c3 b
47. Sinceonelightyearis5 9 1012 miles,Centauriisabout4
50. Eachperson’sshareisequalto
51. First,weestimatethetotalmassofthestarsintheobservableuniverse:
whichisnegative.
Thus,thenumberofhydrogenatomsintheobservableuniverseis
Totalstarmass
Massofasinglehydrogenatom 531 1053 1 67 10 27 318 1080
52.(a)
(b) Answerswillvary.
Patient Weight Height BMI 703 W H 2 Result
A 295lb 5ft10in. 70in. 42 32 obese
B 105lb 5ft6in. 66in. 16 95 underweight
C 220lb 6ft4in. 76in. 26 78 overweight
D 110lb 5ft2in. 62in. 20 12 normal
53. I 15,000 1 0037512n 1
54. Since106 103 103 itwouldtake1000days 2 74yearstospendthemilliondollars.
Since109 103 106 itwouldtake106 1,000,000days 273972yearstospendthebilliondollars.
56.(a)
.Because m n ,wecancancel n factorsof a fromnumeratoranddenominatorandareleftwith
RATIONALEXPONENTSANDRADICALS
1.(a) Usingexponentialnotationwecanwrite 3 5as513
(b) Usingradicalswecanwrite512 as 5.
(c) No.
2.
3. Becausethedenominatorisoftheform
4.
5. No.If
6. No.Forexample,if
(b) 4 64 y 6 4 24 y 4 4 y 2 2 y 4 4 y 2 18.(a) 4 x 2 z 5
22.(a)
46.
63.(a)
(b) Wefindacommonroot:
3 5 3.
73. Firstconvert1135feettomiles.Thisgives1135ft
74.(a) Using f 0 4andsubstituting d 65,weobtain s 30
65 28mi/h. (b) Using f 05andsubstituting s 50,wefind d .Thisgives
2500 15d d 500 3 167feet.
75. Since1day 86,400s,365 25days 31,557,600s.Substituting,weobtain d
76.(a) Substitutingthegivenvaluesweget
(b) Sincethevolumeoftheflowis V A ,thecanaldischargeis17
77.(a)
Sowhen n getslarge,21 n decreasesto1.
(b)
Sowhen n getslarge, 1 2 1 n increasesto1.
P.5 ALGEBRAICEXPRESSIONS
1.(a) 2 x 3 1 2 x 3isapolynomial.(Theconstanttermis notaninteger,butallexponentsareintegers.)
(b) x 2 1 2 3 x x 2 1 2 3 x 12 isnotapolynomialbecausetheexponent 1 2 isnotaninteger.
(c) 1 x 2 4 x 7 isnotapolynomial.(Itisthereciprocalofthepolynomial x 2 4 x 7.)
(d) x 5 7 x 2 x 100isapolynomial.
(e) 3 8 x 6 5 x 3 7 x 3isnotapolynomial.(Itisthecuberootofthepolynomial8 x 6 5 x 3 7 x 3.)
(f) 3 x 4 5 x 2 15 x isapolynomial.(Somecoefficientsarenotintegers,butallexponentsare integers.)
2. Toaddpolynomialsweadd like terms.So
3 x 2
3. Tosubtractpolynomialswesubtract like terms.So
4. WeuseFOILtomultiplytwopolynomials:
5. TheSpecialProductFormulaforthe“squareofasum”is
6. TheSpecialProductFormulaforthe“productofthesumanddifferenceofterms”is
6 x 6 x
7.(a) No;
Yes;
8.(a) Yes;byaSpecialProductFormula,
(b) No, x
x 2 a 2 ,byaSpecialProductFormula.
9. Type:binomial.Terms:5 x 3 and6.Degree:3.
10. Type:trinomial.Terms: 2 x 2 ,5 x ,and 3.Degree:2.
11. Type:monomial.Term: 8.Degree:0.
12. Type:monomial.Terms: 1 2 x 7 .Degree:7.
13. Type:quadrinomial.Terms: x , x 2 , x 3 ,and x 4 .Degree:4.
14. Type:binomial.Terms: 2 x and 3.Degree:1.
15.
SoLHS RHS,thatis,
85.(a) Theheightoftheboxis x ,itswidthis6 2 x ,anditslengthis10 2 x .SinceVolume height width length,we have V x 6 2 x
10 2 x (b) V x 60 32 x
(c) When x 1,thevolumeis V
32,andwhen x 2,thevolumeis
86.(a) Thewidthisthewidthofthelotminusthesetbacksof10feeteach.Thuswidth x 20andlength y 20.Since Area width length,weget A
(b) A x 20 y 20
(c) Forthe100 400lot,thebuildingenvelopehas
30,400.Forthe200 200, lotthebuildingenvelopehas A
200lothasalarger buildingenvelope.
87.(a) A 4000
(b) Rememberthat%meansdivideby100,so2%
88.(a) When
(b)
89.(a) Thedegreeoftheproductisthesumofthedegreesoftheoriginalpolynomials.
(b) Thedegreeofthesumcouldbelowerthaneitherofthedegreesoftheoriginalpolynomials,butisatmostthelargestof thedegreesoftheoriginalpolynomials.
(c) Product:
P.6 FACTORING
1.(a) Thepolynomial2 x 3 3 x 2 10 x hasthreeterms:2 x 3 ,3 x 2 ,and10 x
(b) Thefactor x iscommontoeachterm,so2 x 3 3 x 2
2. Thegreatestcommonfactorintheexpression18 x 3 30 x is6 x ,andtheexpressionfactorsas6 x 3
3. Tofactorthetrinomial x 2 8 x 12welookfortwointegerswhoseproductis12andwhosesumis8.Theseintegersare6 and2,sothetrinomialfactorsas
4. TheSpecialFactoringFormulaforthe“differenceofsquares”is
5. TheSpecialFactoringFormulafora“perfectsquare”is
6. Thegreatestcommonfactorintheexpression4
Startbyfactoringoutthepowerof
56. Startbyfactoringoutthepowerof x withthesmallestexponent,thatis, x 13
Thus, x 53
57. Startbyfactoringoutthepowerof x 2 1withthesmallestexponent,thatis,
Startbyfactoringoutthepowerof
(differenceofsquares)
114.(a) Mowedportion field habitat (b) Usingthedifferenceofsquares,weget
115. Thevolumeoftheshellisthedifferencebetweenthevolumesoftheoutside cylinder(withradius R )andtheinsidecylinder (withradius r ).Thus V
averageradiusis R
2 and2 R
2 istheaveragecircumference(lengthoftherectangularbox), h istheheight,and R r isthethicknessoftherectangularbox.Thus
(b) Basedonthepatterninpart(a),wesuspectthat
P.7
RATIONALEXPRESSIONS
1. Arationalexpressionhastheform P x Q x ,where P and Q arepolynomials.
(a) 3 x x 2 1 isarationalexpression.
(b) x 1 2 x 3 isnotarationalexpression.Arationalexpressionmustbeapolynomialdividedbyapolynomial,andthe numeratoroftheexpressionis x 1,whichisnotapolynomial.
(c) x x 2 1 x 3 x 3 x x 3 isarationalexpression.
2. Tosimplifyarationalexpressionwecancelfactorsthatarecommontothe numerator and denominator.So,theexpression
x 1 x 2
x 3
3. Tomultiplytworationalexpressionswemultiplytheir numerators togetherandmultiplytheir denominators together.So 2 x 1
4.(a)
hasthreeterms.
(b) Theleastcommondenominatorofallthetermsis
5.(a) Yes.Cancelling x 1,wehave
(b) No;
6.(a) Yes, 3 a 3 3 3
(b) No.Wecannot“separate”thedenominatorinthisway;onlythenumerator,asinpart(a).(SeealsoExercise101.)
7. Thedomainof4 x 2 10 x 3isallrealnumbers. 8. Thedomainof x 4 x 3 9 x isallrealnumbers. 9. Since x 3 0wehave x 3.Domain: x
11. Since x 3 0, x 3.Domain;
x 2 x 2
39.
40. x y z x y z x 1 z y xz y 41.
42.
43.
numeratoranddenominatorbythecommondenominatorofboththenumerator anddenominator,inthiscase x 2 y 2 :
76. Incalculusitisnecessarytoeliminatethe h inthedenominator,andwedothisbyrationalizingthenumerator:
99.(a) R
(b) Substituting R1 10ohmsand R2
100.(a) Theaveragecost A
(b)
101.
Fromthetable,weseethattheexpression x 2 9 x 3 approaches6as x approaches3.Wesimplifytheexpression: x 2
3.Clearlyas x approaches3, x 3approaches6.Thisexplainstheresultinthe table.
102. No,squaring 2 x changesitsvaluebyafactorof 2 x .
103. Answerswillvary.
AlgebraicError Counterexample
104.(a) 5 a 5 5 5 a 5 1 a 5 ,sothestatementistrue.
(b) Thisstatementisfalse.Forexample,take x 5and y 2.ThenLHS
RHS x y 5 2 ,and2 5 2
(c) Thisstatementisfalse.Forexample,take x 0and y 1.ThenLHS x x y 0 0
1 0,while RHS 1 1 y 1 1 1 1 2 ,and0 1 2 .
(d) Thisstatementisfalse.Forexample,take x 1and y 1.ThenLHS
RHS 2a 2b 2 2 1,and2 1.
(e) Thisstatementistrue:
(f) Thisstatementistrue:
Itappearsthatthesmallestpossiblevalueof
(b) Because x 0,wecanmultiplybothsidesby x andpreservetheinequality:
x 1
2
0.Thelaststatementistrueforall x 0,andbecauseeachstepis reversible,wehaveshownthat x 1 x 2forall x 0.
P.8 SOLVINGBASICEQUATIONS
1. Substituting x 3intheequation4 x 2 10makestheequationtrue,sothenumber3isa solution oftheequation.
2. Subtracting4frombothsidesofthegivenequation,3x 4 10,weobtain3 x 4 4 10 4 3 x 6.Multiplying by 1 3 ,wehave 1 3 3 x 1 3 6 x 2,sothesolutionis x 2.
3.(a) x 2 2 x 10isequivalentto 5 2 x 10 0,soitisalinearequation.
(b) 2 x 2 x 1isnotlinearbecauseitcontainstheterm 2 x ,amultipleofthereciprocalofthevariable.
(c) x 7 5 3 x 4 x 2 0,soitislinear.
4.(a) x x 1 6 x 2 x 6isnotlinearbecauseitcontainsthesquareofthevariable.
(b) x 2 x isnotlinearbecauseitcontainsthesquarerootof x 2.
(c) 3 x 2 2 x 1 0isnotlinearbecauseitcontainsamultipleofthesquare ofthevariable.
5.(a) Thisistrue:If a b ,then a x b x
(b) Thisisfalse,becausethenumbercouldbezero.However,itistruethatmultiplyingeachsideofanequationbya nonzero numberalwaysgivesanequivalentequation.
(c) Thisisfalse.Forexample, 5 5isfalse,but 52 52 istrue.
6. Tosolvetheequation x 3 125wetakethe cube rootofeachside.Sothesolutionis x 3 125 5.
7.(a) When x 2,LHS 4 2 7
18 3 21.SinceLHS RHS, x 2isnotasolution.
(b) When x 2,LHS 4 2 7 8 7 15andRHS 9 2 3 18 3 15.SinceLHS RHS, x 2isa solution.
8.(a) When x 1,LHS 2 5 1 2 5 7andRHS
(b) When x 1,LHS 2 5 1 2 5 3andRHS 8
9.(a) When x 2,LHS 1 [2
7.SinceLHS RHS, x 1isasolution.
1 9.SinceLHS RHS, x 1isnotasolution.
8 8
0.Since LHS RHS, x 2isasolution.
(b) When x 4LHS 1
16 10 6. SinceLHS RHS, x 4isnotasolution.
10.(a) When x 2,LHS
1andRHS 1.SinceLHS RHS, x 2isasolution.
(b) When x 4theexpression 1 4 4 isnotdefined,so x 4isnotasolution.
11.(a) When x
(b) When x 8LHS
12.(a) When x 4,LHS
(b) When x 8,LHS
5.SinceLHS 1, x 1isnotasolution.
RHS.So x 8isasolution.
4.SinceLHS RHS, x 4isasolution.
8isnota solution.
13.(a) When x 0,LHS 0
(b) When x
14.(a) When
(b) When
RHS.So x b 2 isasolution.
Butsubstituting x 2intotheoriginalequationdoesnotwork,sincewecannotdivideby0.Thus thereisnosolution.
53. 3 x 4 1 x 6 x 12 x 2 4 x 3
4 x 16 x 4.Butsubstituting x 4intotheoriginalequationdoesnotwork,sincewecannotdivideby0.Thus,thereis nosolution.
54. 1 x 2 2 x
1.Thisisanidentityfor x 0and x 1 2 ,sothesolutionsare allrealnumbersexcept0and 1 2 .
55. x 2 25 x 5
56. 3 x 2 48 x 2 16 x 4
57. 5 x 2 15 x 2 3 x 3
58. x 2 1000 x 1000 1010
59. 8 x 2 64
60. 5 x 2 125 0 5 x 2 25 0 x 2 25 x 5
61. x 2 16 0 x 2 16whichhasnorealsolution.
62.
63.
101.(a) Theshrinkagefactorwhen
00055 12
(b) Substituting
102. Substituting C 3600weget3600
manufacture840toytrucks.
103.(a) Solvingfor when P 10,000weget10,000
Solvingfor
104. Substituting F
Thatis, x
107. Whenwemultipliedby x ,weintroduced x 0asasolution.Whenwedividedby x 1,wearereallydividingby0,since x 1 x 1 0.
P.9
MODELINGWITHEQUATIONS
1. Anequationmodelingarealworldsituationcanbeusedtohelpusunderstandarealworldproblemusingmathematical methods.Wetranslaterealworldideasintothelanguageofalgebratoconstructourmodel,andtranslateourmathematical resultsbackintorealworldideasinordertointerpretourfindings.
2. Intheformula I Prt forsimpleinterest, P standsfor principal, r for interestrate,and t for time(inyears)
3.(a) Asquareofside x hasarea A x 2
(b) Arectangleoflength l andwidth hasarea A l .
(c) Acircleofradius r hasarea A r 2
4. Balsamicvinegarcontains5%aceticacid,soa32ouncebottleofbalsamicvinegarcontains32 5% 32 5 100 1 6ounces ofaceticacid.
5. Apainterpaintsawallin x hours,sothefractionofthewallshepaintsinonehouris 1wall x hours 1 x
6. Solving d rt for r ,wefind d t rt t r d t .Solving d rt for t ,wefind d r rt r t d r
7. If n isthefirstinteger,then n 1isthemiddleinteger,and n 2isthethirdinteger.Sothesumofthethreeconsecutive integersis n n 1 n
8. If n isthemiddleinteger,then n 1isthefirstinteger,and n 1isthethirdinteger.Sothesumofthethreeconsecutive integersis
9. If n isthefirsteveninteger,then n 2isthesecondevenintegerand n 4isthethird.Sothesumofthreeconsecutive evenintegersis n
10. If n isthefirstinteger,thenthenextintegeris n 1.Thesumoftheirsquaresis
11. If s isthethirdtestscore,thensincetheothertestscoresare78and82,theaverageofthethreetestscoresis 78 82 s 3
12. If q isthefourthquizscore,thensincetheotherquizscoresare8,8,and8,the averageofthefourquizscoresis 8 8 8 q 4
13. If x dollarsareinvestedat2 1 2 %simpleinterest,thenthefirstyearyouwillreceive0025 x dollarsininterest.
14. If n isthenumberofmonthstheapartmentisrented,andeachmonththerentis$945,thenthetotalrentpaidis945n .
15. Since isthewidthoftherectangle,thelengthisthreetimesthewidth,or4 .Then area length width
16. Since isthewidthoftherectangle,thelengthis 6.Theperimeteris
17. If d isthegivendistance,inmiles,anddistance rate time,wehavetime distance rate d 55 .
18. Sincedistance rate timewehavedistance s 45min 1h 60min 3 4 s mi.
19. If x isthequantityofpurewateradded,themixturewillcontain25ozofsaltand3 x gallonsofwater.Thusthe concentrationis 25 3 x
20. If p isthenumberofpenniesinthepurse,thenthenumberofnickelsis2 p ,thenumberofdimesis4 2 p ,and thenumberofquartersis
4.Thusthevalue(incents)ofthechangeinthepurseis
21. If d isthenumberofdaysand m thenumberofmiles,thenthecostofarentalis C
3and C
283,sowesolvefor
truckwasdriven220miles.
22. Thecostofspeakingfor m minutesonthisplanis C
and C 120 50,sowesolve100
m 250 82 332.Therefore,thetouristused332minutesoftalkinthemonthofJune.
23. If x isthestudent’sscoreontheirfinalexam,thenbecausethefinalcountstwiceasmuchaseachmidterm,theaverage scoreis
x
.Fortheaveragetobe80%,wemusthave
86.Sothestudentscored86%ontheirfinalexam.
24. Sixstudentsscored100andthreestudentsscored60.Let x betheaveragescoreoftheremaining25 6 3 16students.
Becausetheoverallaverageis84% 0 84,wehave
825.Thus,theremaining16students’averagescorewas825%.
25. Let m betheamountinvestedat2 1 2 %.Then12,000 m istheamountinvestedat3%.
Sincethetotalinterestisequaltotheinterestearnedat2 5%plustheinterestearnedat3%,wehave
$8400isinvestedat2 1 2 %and12,000 8400 $3600isinvestedat3%.
26. Let m betheamountinvestedat5%.Then8000 m isthetotalamountinvested.Thus 4%ofthetotalinvestment interestearnedat3 1 2 % interestearnedat5%
4000.Thus,$4000mustbe investedat5%.
27. Usingtheformula I Prt andsolvingfor r ,weget262 50 3500 r 1 r 262 5 3500 0 075or7 5%.
28. If$3000isinvestedataninterestrate a %,then$5000isinvestedat a 1 2 %,so,rememberingthat a isexpressedasa percentage,thetotalinterestis I 3000 a
a
25.Sincethetotalinterest is$265,wehave265 80a 25 80a 240 a 3.Thus,$3000isinvestedat3%interest.
29. Let x bethemonthlysalary.Sinceannualsalary 12 monthlysalary Christmasbonus,wehave 180,100 12 x 7300 172,800 12 x x 14,400.Themonthlysalaryis$14,400.
30. Let s betheassistant’sannualsalary.Thentheforeman’sannualsalaryis1 15s .Theirtotalincomeisthesumoftheir salaries,so s 115s 113,305 215s 113,305 s 52,700.Thus,theassistant’sannualsalaryis$52,700.
31. Let x betheovertimehoursworked.Sincegrosspay regularsalary overtimepay,weobtaintheequation 814
x x 16650 2775 6.Thus,thelab technicianworked6hoursofovertime.
32. Let x bethenumberofhoursworkedbytheplumber.Thenthecabinetmakerworksfor9 x hours.Thetotallaborchargeis thesumoftheircharges,so2610 150 x 80 9 x 2610 150 x 720 x x 2610 870 3.Thus,theplumberworks for3hoursandthecabinetmakerworksfor3 9 27hours.
33. Allagesareintermsofthedaughter’sage7yearsago.Let y beageofthedaughter7yearsago.Then11 y istheageofthe moviestar7yearsago.Today,thedaughteris y 7,andthemoviestaris11 y 7.Butthemoviestarisalso4timeshis daughter’sagetoday.So4
3.Thus,todaythemoviestaris 11 3 7 40yearsold.
34. Let h benumberofhomerunsBabeRuthhit.Then h 41isthenumberofhomerunsthatHankAaronhit.So 1469 h h 41 1428 2h h 714.ThusBabeRuthhit714homeruns.
35. Let n bethenumberofnickels.Thentherearealso n dimesand n quarters.Thetotalvalueofthecoinsinthepurseisthe sumofthevaluesofnickels,dimes,andquarters,so2 80
pursecontains7nickels,7dimes,and7quarters.
36. Let q bethenumberofquarters.Then2q isthenumberofdimes,and2q 5isthenumberofnickels.Thus3 00 value ofnickels valueofdimes valueofquarters,so 3
Thusyouhave5quarters,2
5
10dimes,and2
5
5 15nickels.
37. Let l bethelengthofthegarden.Sincearea width length,weobtaintheequation1125
45ft.So thegardenis45feetlong.
38. Let bethewidthofthepasture.Thenthelengthofthepastureis3 .Sincearea length widthwehave 132,300
210.Thusthewidthofthepastureis210feet.
39. Let bethewidthofthebuildinglot.Thenthelengthofthebuildinglotis5 .Sinceahalfacreis 1 2 43,560 21,780 andareaislengthtimeswidth,wehave21,780
66.Thusthewidthofthe buildinglotis66feetandthelengthofthebuildinglotis5 66 330feet.
40. Let x bethelengthofasideofthesquareplot.Asshowninthefigure, areaoftheplot areaofthebuilding areaoftheparkinglot.Thus, x 2 60 40 12,000 2,400 12,000 14,400 x 120.Sotheplotof landmeasures120feetby120feet. x x
41. Theshadedareaisthesumoftheareaofasquareandtheareaofatriangle.So A y 2
.Wearegiven thattheareais120in2 ,so120
42. Firstwewriteaformulafortheareaofthefigureintermsof x .Region A has dimensions10cmand x cmandregion B hasdimensions6cmand x cm.Sothe shadedregionhasarea 10 x 6 x 16 x cm2 .Wearegiventhatthisisequal to144cm2 ,so144 16 x x 144 16 9cm.
43. Let x bethewidthofthestrip.Thenthelengthofthematis20 2 x ,andthewidthofthematis15 2 x .Nowthe perimeteristwicethelengthplustwicethewidth,so102 2 20 2 x 2 15
102 70 8 x 32 8 x x 4.Thusthestripofmatis4incheswide.
44. Let x bethewidthofthestrip.Thenthewidthoftheposteris100 2 x anditslengthis140 2 x .Theperimeterofthe printedareais2 100 2 140 480,andtheperimeteroftheposteris2 100
.Nowweusethe factthattheperimeteroftheposteris1 1 2 timestheperimeteroftheprintedarea:2
480 8 x 720 8 x 240 x 30.Theblankstripisthus30cmwide.
45. Let x bethelengthoftheperson’sshadow,inmeters.Usingsimilartriangles, 10 x 6 x 2 20 2 x 6 x 4 x 20 x 5.Thustheperson’sshadowis5meterslong.
46. Let x betheheightofthetalltree.Hereweusethepropertythatcorresponding sidesinsimilartrianglesareproportional.Thebaseofthesimilartrianglesstartsat eyelevelofthewoodcutter,5feet.Thusweobtaintheproportion x 5 15
treeis95feettall.
47. Let x betheamount(inmL)of60%acidsolutiontobeused.Then300 x mLof30%solutionwouldhavetobeusedto yieldatotalof300mLofsolution.
Thusthetotalamountofpureacidusedis0
So200mLof60%acidsolutionmustbemixedwith100mLof30%solutiontoget300mLof50%acidsolution.
48. Theamountofpureacidintheoriginalsolutionis300 50% 150.Let x bethenumberofmLofpureacidadded.Then thefinalvolumeofsolutionis300 x .Becauseitsconcentrationistobe60%,wemusthave
49. Let x bethenumberofgramsofsilveradded.Theweightoftheringsis5 18g
mustbeaddedtogettherequiredmixture.
50. Let x bethenumberoflitersofwatertobeboiledoff.Theresultwillcontain6 x liters.
51. Let x bethenumberoflitersofcoolantremovedandreplacedbywater.
mustberemovedandreplacedbywater.
52. Let x bethenumberofgallonsof2%bleachremovedfromthetank.Thisisalsothenumberofgallonsofpurebleach addedtomakethe5%mixture.
06gallonsneedtoremoved andreplacedwithpurebleach.
53. Let c betheconcentrationoffruitjuiceinthecheaperbrand.Thenewmixtureconsistsof650mLoftheoriginalfruitpunch and100mLofthecheaperfruitpunch. OriginalFruitPunch CheaperFruitPunch Mixture
35%fruitjuice.
54. Let x bethenumberofouncesof$300ozteaThen80 x isthenumberofouncesof$2
75oztea.
48.Themixtureuses 48ouncesof$3 00ozteaand80 48
32ouncesof$2
55. Let t bethetimeinminutesitwouldtaketowashthecarifthefriendsworkedtogether.Friend1washes 1 25 ofthecarper minute,whileFriend2washes 1 35 ofthecarperminute.Thesumofthese fractionsisequaltothefractionofthejobthey candoworkingtogether,sowehave
t 14 35 60 minutes,or14minutes35seconds
56. Let t bethetime,inminutes,ittakesthelandscapertomowthelawn.Sincetheassistantishalfasfast,itwouldtakethem 2t minutestomowthelawnalone.Thus,
15.Thus,itwouldthe assistant2 15 30minutestomowthelawnalone.
57. Let t bethenumberofhoursitwouldtakeyourfriendtopaintahousealone.Thenworkingtogether,ittakes 2 3 t hours.
Becauseittakesyou7hours,wehave
wouldtakeyourfriend3 5htopaintahousealone.
58. Let h bethetime,inhours,tofilltheswimmingpoolusingthesmallerhosealone. Sincethelargerhosetakes20%less time,ittakes0 8h tofillthepoolalone.Thus16
59. Let t bethetimeinhoursthatthecommuterspentonthetrain.Then 11 2 t isthetimeinhoursthatcommuterspentonthe bus.Weconstructatable:
Thetotaldistancetraveledisthesumofthedistancestraveledbybusandbytrain,so300
60. Let r bethespeedoftheslowercyclist,inmi/h.Thenthespeedofthefastercyclistis2r
Whentheymeet,theywillhavetraveledacombinedtotalof90miles,so2
15.Thespeed oftheslowercyclistis15mi/h,whilethespeedofthefastercyclistis2 15
61. Let r bethespeedoftheplanefromMontrealtoLosAngeles.Then r
30mi/h.
1 20r isthespeedoftheplanefromLos AngelestoMontreal.
Thetotaltimeisthesumofthetimeseachway,so9
ataspeedof500mi/honthetripfromMontrealtoLosAngeles.
62. Let x bethespeedofthecarinmi/h.Sinceamilecontains5280ftandanhourcontains3600s,1mi/h
22 15 ft/s. Thetruckistravelingat50 22 15 220 3 ft/s.Soin6seconds,thetrucktravels6 220 3 440feet.Thusthebackend ofthecarmusttravelthelengthofthecar,thelengthofthetruck,andthe440feetin6seconds,soitsspeedmustbe 1430440 6 242 3 ft/s.Convertingtomi/h,wehavethatthespeedofthecaris 242 3 15 22 55mi/h.
63. Let x bethedistancefromthefulcrumtowherethe125poundfriendsits.Thensubstitutingtheknownvaluesintothe formulagiven,wehave100 8 125 x 800 125 x x 64.Sothe125poundfriendshouldsit64feetfromthe fulcrum.
64. Let bethelargestweightthatcanbehung.Inthisexercise,theedgeofthebuildingactsasthefulcrum,sothe240lb manissitting25feetfromthefulcrum.Thensubstitutingtheknownvalues intotheformulagiveninExercise63,wehave 240
25
1200.Therefore,1200poundsisthelargestweightthatcanbehung.
65. Let l bethelengthofthelotinfeet.Thenthelengthofthediagonalis l 10.We applythePythagoreanTheoremwith thehypotenuseasthediagonal.So
Thusthelengthofthelotis120feet.
66. Let r betheradiusoftherunningtrack.Therunningtrackconsistsoftwosemicirclesandtwostraightsections110yards long,sowegettheequation2
03.Thustheradiusofthesemicircleis about35yards.
67. Let h betheheightinfeetofthestructure.Thestructureiscomposedofarightcylinderwithradius10andheight 2 3 h anda conewithbaseradius10andheight 1 3 h .Usingtheformulasforthevolumeofacylinderandthatofacone,weobtain the equation1400
(multiplybothsides by 9 100 ) 126 7
18.Thustheheightofthestructureis18feet.
68. Let h betheheightofthebreak,infeet.Thentheportionofthebambooabovethe breakis10 h .ApplyingthePythagoreanTheorem,weobtain
h 91 20 4 55.Thusthebreakis4 55ftabovetheground.
69. Answerswillvary.
CHAPTERPREVIEW
1.(a) Sincethereareinitially250tabletsandthepatienttakes2tabletsperday,thenumberoftablets T thatareleftinthe bottleafter x daysis T 250 2 x
(b) After30days,thereare250 2 30 190tabletsleft.
(c) Weset T 0andsolve: T
x 125.Thepatientwillrunoutafter125days.
2.(a) Thetotalcostis$11percalzoneplusthe$7deliverycharge,so C 11 x 7.
(b) Fourcalzoneswouldcost11 4 7 $51.
(c) Wesolve C 11 x 7 40 11 x 33 x 3.Youcanorderthreecalzones.
3.(a) 16isrational.Itisaninteger,andmoreprecisely,anaturalnumber.
(b) 16isrational.Itisaninteger,butbecauseitisnegative,itisnotanaturalnumber.
(c) 16 4isrational.Itisaninteger,andmoreprecisely,anaturalnumber.
(d) 2isirrational.
(e) 8 3 isrational,butisneitheranaturalnumbernoraninteger.
(f) 8 2 4isrational.Itisaninteger,butbecauseitisnegative,itisnotanaturalnumber.
4.(a) 5isrational.Itisaninteger,butnotanaturalnumber.
(b) 25 6 isrational,butisneitheranintegernoranaturalnumber.
(c) 25 5isrational,anaturalnumber,andaninteger.
(d) 3 isirrational.
(e) 24 16 3 2 isrational,butisneitheranaturalnumbernoraninteger. (f) 1020 isrational,anaturalnumber,andaninteger.
5. CommutativePropertyforaddition. 6. CommutativePropertyformultiplication.
7. DistributiveProperty. 8. DistributiveProperty.
35.(a)
37.(a)
43. 78,250,000,000 7 825 1010
44.
45. ab c
165
46.
x
92. 2 x x 2 9 isdefinedwhenever x 2 9 0 x 2 9 x 3,soitsdomainis x x 3and x 3
93. x x 2 3 x 4 isdefinedwhenever x 0(sothat x isdefined)and x 2 3 x 4 x 1 x 4 0 x 1and x 4.Thus,itsdomainis x x 0and x 4.
94. x 3 x 2 4 x 4 isdefinedwhenever x 3 0 x
95. Thisstatementisfalse.Forexample,take
RHS x 3 y 3 13 13 1 1
98. Thisstatementisfalse.Forexample,take
99. Thisstatementisfalse.Forexample,take
lastequationisnevertrue,thereisnorealsolutiontotheoriginalequation.
127. Let x bethenumberofpoundsofraisins.Thenthenumberofpoundsofnutsis50 x Raisins
128. Let t bethenumberofhoursthatthedistrictsupervisordrives.Thenthestoremanagerdrivesfor t 1 4 hours.
Whentheypasseachother,theywillhavetraveledatotalof160miles.So45
2.Sincethesupervisorleavesat2:00 P M .andtravelsfor2hours,theypasseachotherat4:00 P M
129. Let x betheamountinvestedintheaccountearning1 5%interest.Thentheamountinvestedintheaccountearning2 5%is 7000 x
x 5475.Thus,$5475isinvestedintheaccountearning1 5%interestand$1525isinvestedintheaccountearning 25%interest.
130. Theamountofinterestcurrentlyearnedis6000 0 03
$180peryear.Ifatotalof$300isdesired,another$120ininterest mustbeearnedatarateof125%peryear.Iftheadditionalamountinvestedis x ,wehave00125 x 120 x 9600. Thus,anadditional$9600mustbeinvestedat1 25%simpleinteresttoearnatotalof$300interestperyear.
131. Let t bethetimeitwouldtaketheinteriordecoratortopaintalivingroomiftheyworkalone.Itwouldtaketheassistant 2t hoursalone,anditwouldtaketheapprentice3t hoursalone.Thus,thedecoratordoes 1 t ofthejobperhour,theassistant does 1 2t ofthejobperhour,andtheapprenticedoes 1 3t ofthejobperhour.So
6t 11 t 11 6 .Thus,itwouldtakethedecorator1hour50minutestopaintthelivingroom alone.
132. Let bewidthofthepool.Thenthelengthofthepoolis2 ,anditsvolumeis8
2 529 23.Since 0,werejectthenegativevalue.Thepoolis23feetwide,2 23 46feetlong,and 8feetdeep.
CHAPTERPTEST
1.(a) Thecostis C 9 1 5 x
(b) Therearefourextratoppings,so x 4and C 9 1 5 4 $15.
(c) Wehave C 19 5 9 1 5 x 1 5 x 10 5 x 7.Thus,a$19 50pizzahas7toppings.
2.(a) 5isrational.Itisaninteger,andmoreprecisely,anaturalnumber.
(b) 5isirrational.
(c) 9 3 3isrational,anditisaninteger.
(d) 1,000,000isrational,anditisaninteger.
3.(a) A B 0 1 5
(b) A B 2 0 1 2 1 3 5 7
4.(a)
5.(a)
6.(a)
7.(a)
8.(a)
.(Wetakethepositiverootbecause c representsthespeedoflight,whichispositive.)
15. Let t bethetime(inhours)ittookthetruckertodrivefromAmitytoBelleville. Then44 t isthetimeittookthetrucker todrivefromBellevilletoAmity.SincethedistancefromAmitytoBellevilleequalsthedistancefromBellevilletoAmity, wehave50
120mi.
FOCUSONMODELINGMakingOptimalDecisions
1.(a) Thetotalcostis
(b) Inthiscasethecostis
(d) Thecostisthesamewhen C 1 C 2 areequal.So5800 265n 575n
2.(a) ThecostofPlan1is3
ThecostofPlan2is3
daily cost
daily cost
costper mile
(b) When x 400,Plan1costs195 0 15 400 $255andPlan2costs$270,soPlan1ischeaper.When x 800, Plan1costs195 0 15 800 $315andPlan2costs$270,soPlan2ischeaper.
(c) Thecostisthesamewhen195 0 15 x 270 0 15 75 x x 500.Sobothplanscost$270whenyoudrive 500miles.
3.(a) Thetotalcostis setup cost
(b) Therevenueis
priceper tire
costper tire
number oftires
number oftires
.So R 49 x
.So C 8000 22 x
(c) Profit Revenue Cost.So P R C 49 x 8000 22 x 27 x 8000.
(d) Breakeveniswhenprofitiszero.Thus27 x 8000 0 27 x 8000 x 296 3.Sotheyneedtosellatleast 297tirestobreakeven.
4.(a) Option1: Inthisoptionthewidthisconstantat100.Let x betheincreaseinlength.Thentheadditionalareais
width
increase inlength
100 x .Thecostisthesumofthecostsofmovingtheoldfence,andofinstallingthe
newone.Thecostofmovingis$6 100 $600andthecostofinstallationis2 10 x 20 x ,sothetotalcostis
C 20 x 600.Solvingfor x ,weget C 20 x 600 20 x C 600 x C 600 20 .Substitutinginthearea
A
Option2: Inthisoptionthelengthisconstantat180.Let y betheincreaseinthewidth.Thentheadditionalareais
length
(b)
increase inwidth
180 y .Thecostofmovingtheoldfenceis6 180 $1080andthecostofinstallingthenew
oneis2 10 y 20 x ,sothetotalcostis C 20 y 1080.Solvingfor y ,weget C 20 y 1080 20 y C 1080
y C 1080 20 .Substitutingintheareawehave A 2 180 C 1080 20 9 C 1080 9C 9,720.
Cost C Areagain A1 fromOption1 Areagain A2 fromOption2
$1100 2,500ft2 180ft2
$1200 3,000ft2 1,080ft2
$1500 4,500ft2 3,780ft2
$2000 7,000ft2 8,280ft2
(c) Ifthefarmerhasonly$1200,Option1givesthegreatestgain.Ifthefarmer hasonly$2000,Option2givesthegreatest gain.
5.(a) Design1isasquareandtheperimeterofasquareisfourtimesthelengthofa side.24 4 x ,soeachsideis x 6feet long.Thustheareais62 36ft2
Design2isacirclewithperimeter2r andarea r 2 .Thuswemustsolve2r 24 r 12 .Thus,theareais
12 2 144 458ft2 .Design2givesthelargestarea.
(b) InDesign1,thecostis$3timestheperimeter p ,so120 3 p andtheperimeteris40feet.Bypart(a),eachsideisthen 40 4 10feetlong.Sotheareais102 100ft2
InDesign2,thecostis$4timestheperimeter p .Becausetheperimeteris2r ,weget120 4 2r so r 120 8 15 .Theareais r 2 15 2 225 716ft2 .Design1givesthelargestarea.
6.(a) Plan1:Tomatoeseveryyear.Profit acres Revenue cost 100 1600 300 130,000.Thenfor n yearsthe profitis P1 130,000n
(b) Plan2:Soybeansfollowedbytomatoes.TheprofitfortwoyearsisProfit acres
soybean revenue
tomato revenue
100 1200 1600 280,000.Rememberthatnofertilizeris neededinthisplan.Thenfor2k years,theprofitis P2 280,000k
(c) When n 10, P1 130,000 10 1,300,000.Since2k 10when k 5, P2 280,000 5 1,400,000.SoPlanB ismoreprofitable.
60 12 1
$72.
If3 7GBareused,PlanAcosts25
60 27 1 $87.
If4 9GBareused,PlanAcosts25
60 39 1 $99.
$103,PlanBcosts40
50,andPlanCcosts
(d)(i) Weset C A C B 20 x 5 15 x 25 5 x 20 x 4.PlansAandBcostthesamewhen4GBareused.
(ii) Weset
4 5.PlansAandCcostthesamewhen4 5GBare used.
(iii) Weset
5.PlansBandCcostthesamewhen5GBare used.
8.(a) Inthisplan,CompanyAgets$3 2millionandCompanyBgets$3 2million.CompanyA’sinvestmentis$1 4million, sotheymakeaprofitof32 14 $18million.CompanyB’sinvestmentis$26million,sotheymakeaprofitof 3 2 2 6 $0 6million.SoCompanyAmakesthreetimestheprofitthatCompanyBdoes,whichisnotfair.
(b) Theoriginalinvestmentis1 4 2 6 $4million.Soaftergivingtheoriginalinvestmentback,theythensharethe profitof$24million.Soeachgetsanadditional$12million.SoCompanyAgetsatotalof14 12 $26million andCompanyBgets2 6 1 2 $3 8million.SoeventhoughCompanyBinvestsmore,theymakethesameprofitas CompanyA,whichisnotfair.
(c) Theoriginalinvestmentis$4million,soCompanyAgets 1 4 4 6 4 $2 24millionandCompanyBgets
2 6 4 6 4 $4 16million.Thisseemsthefairest.